Two-grid finite volume element methods for semilinear parabolic problems

Two-grid finite volume element methods, based on two linear conforming finite element spaces on one coarse grid and one fine grid, are presented and studied for two-dimensional semilinear parabolic problems. With the proposed techniques, solving the nonsymmetric and nonlinear system on the fine space is reduced to solving a symmetric and linear system on the fine space and solving the nonsymmetric and nonlinear system on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. It is proved that the coarse grid can be much coarser than the fine grid. As a result, solving such a large class of semilinear parabolic problems will not be much more difficult than solving one single linearized equation. In the end a numerical example is presented to validate the usefulness and efficiency of the method.